# Warwick Mathematics Institute Events

Click on a title to view the abstract!

## Upcoming Seminars

• Geometry and Topology on 01 December 2022 at 14:00 in B3.02

Speaker: Koji Fujiwara (Kyoto University)

Title: The rates of growth in a hyperbolic group

Abstract: I discuss the set of rates of growth of a finitely generated group with respect to all its finite generating sets. In a joint work with Sela, for a hyperbolic group, we showed that the set is well-ordered, and that each number can be the rate of growth of at most finitely many generating sets up to automorphism of the group. I may discuss its generalization to acylindrically hyperbolic groups.

• Probability Seminar on 01 December 2022 at 16:00 in MS.04 and online here

Speaker: Sunil Chhita (University of Durham)

Title: Domino Shuffle and Matrix Refactorizations

Abstract: This talk is motivated by computing correlations for domino tilings of the Aztec diamond. It is inspired by two of the three distinct methods that have recently been used in the simplest case of a doubly periodic weighting, that is the two-periodic Aztec diamond. This model is of particular probabilistic interest due to being one of the few models having a boundary between polynomially and exponentially decaying macroscopic regions in the limit. One of the methods to compute correlations, powered by the domino shuffle, involves inverting the Kasteleyn matrix giving correlations through the local statistics formula. Another of the methods, driven by a Wiener-Hopf factorization for two- by-two matrix valued functions, involves the Eynard-Mehta theorem. For arbitrary weights the Wiener-Hopf factorization can be replaced by an LU- and UL-decomposition, based on a matrix refactorization, for the product of the transition matrices. In this talk, we present results to say that the evolution of the face weights under the domino shuffle and the matrix refactorization is the same. This is based on joint work with Maurice Duits (Royal Institute of Technology KTH).

• Applied Mathematics on 02 December 2022 at 12:00 in B3.02

Speaker: Katherine Kamal (Cambridge)

Title: The microhydrodynamics of ultra-thin nanoparticles: modelling to predict the "unseen"

Abstract: Graphene nanoparticles are ubiquitous, used in everything from the design of more robust extreme weather-resistance spacecraft to flexible-electronics tracks. Made from just a few atomic layers, the instantaneous dynamics of these plate-like particles in flowing liquids are, experimentally, practically inaccessible. We study theoretically and computationally the microhydrodynamics of dilute suspensions of graphene in a simple viscous shear flow field. In the infinite Péclet number limit, a rigid platelet with the interfacial hydrodynamic slip properties of graphene does not follow the periodic rotations predicted for classical colloidal particles but aligns itself at a slight inclination angle with respect to the flow. This unexpected result is due to the hydrodynamic slip reducing the tangential stress at the graphene-liquid surface. By analysing the Fokker-Plank equation for the orientational distribution function for decreasing Péclet numbers, we explore how hydrodynamic slip affects the particle’s orientation and effective viscosity. We find that hydrodynamic slip can dramatically change the average particle’s orientation and effective viscosity. For example, the effective viscosity of a dilute suspension of graphene platelets is predicted to be smaller than the base fluids under certain flow conditions for typical slip length values.

• Combinatorics on 02 December 2022 at 14:00 in B3.02

Speaker: Bernd Schulze (University of Lancaster)

Title: Geometric Rigidity Theory and Applications

Abstract: In the last two decades or so the subject has become particularly active, drawing on diverse areas of mathematics, and engaging with a growing range of modern applications, such as Engineering, Robotics, Computer-Aided-Design, Molecular Dynamics, and Materials Science.

In the first part of the talk, I will give an introduction to Geometric Rigidity Theory, concentrating on some key combinatorial results and problems for bar-joint frameworks, but also describing how these have been extended to some other types of frameworks.

Since many real-world structures are symmetric, a major recent research direction in the field is to study the impact of symmetry on the rigidity and flexibility of bar-joint frameworks. I will show how group representation theory can be used to reveal hidden' infinitesimal motions and states of self-stress in symmetric frameworks that cannot be detected with the standard non-symmetric counts. Finally, I will show how these symmetry-based methods can be used as a design tool for gridshell structures. This is recent joint work with William Baker, Arek Mazurek and Cameron Millar.

• Colloquium on 02 December 2022 at 16:00 in B3.02

Speaker: Tara Brendle (Glasgow)

Title: Twists and trivializations: encoding symmetries of manifolds

Abstract: The classification of 2-manifolds in the first half of the 20th century was a landmark achievement in mathematics, as was the more recent (and more complicated) classification of 3-manifolds completed by Perelman. The story does not end with classification, however: there is a rich theory of symmetries of manifolds, encoded in their mapping class groups. In this talk we will explore some aspects of mapping class groups in dimensions 2 and 3, with a focus on illustrative examples.

• Number Theory on 05 December 2022 at 15:00 in B3.02

Speaker: Istvan Kolossvary (St Andrews)

Title: Distance between natural numbers based on their prime signature

Abstract: One can define different metrics between natural numbers based on their unique prime signature. Fixing such a metric, we are interested in the asymptotic growth rate of the arithmetic function L(N) which tabulates the cumulative sum of distances between consecutive natural numbers up to N. In particular, choosing the maximum norm, we will show that the limit of L(N)/N exists and is equal to the expected value of a certain random variable. We also demonstrate that prime gaps exhibit a richer structure than on the traditional number line and pose a number of problems. Joint work with Istvan B. Kolossvary.

• Algebra on 05 December 2022 at 17:00 in B3.02

Speaker: Ana Retegan (University of Birmingham)

Title: TBC

Abstract: TBC

• Partial Differential Equations and their Applications on 06 December 2022 at 12:00 in B3.02

Speaker: Antonio Esposito (University of Oxford)

Title: TBA

Abstract: TBA

• Geometry and Topology on 08 December 2022 at 14:00 in B3.02

Speaker: Ric Wade (University of Oxford)

Title: Aut-invariant quasimorphisms on groups

Abstract: For a large class of groups, we exhibit an infinite-dimensional space of homogeneous quasimorphisms that are invariant under the action of the automorphism group. This class includes non-elementary hyperbolic groups, infinitely-ended finitely generated groups, some relatively hyperbolic groups, and a class of graph products of groups that includes all right-angled Artin and Coxeter groups that are not virtually abelian. Joint work with Francesco Fournier-Facio.

• Combinatorics on 09 December 2022 at 14:00 in B3.02

Speaker: Zoltán Vidnyánszky (Eötvös Loránd University)

Title: Borel combinatorics, the LOCAL model and complexity

Abstract: In the first part of the talk, I will give an overview of the field of Borel combinatorics and its recently uncovered connections to the LOCAL model of distributed computing. Then, I will discuss complexity related aspects of the field. Namely, I will consider the question of how hard it is to decide the existence of Borel homomorphisms from a Borel structure to a given finite structure.

• Colloquium on 09 December 2022 at 16:00 in B3.02

Speaker: John Baez (UC Riverside)

Title: Category Theory in Epidemiology

Abstract: Category theory provides a general framework for building models of dynamical systems. We explain this framework and illustrate it with the example of "stock and flow diagrams". These diagrams are widely used for simulations in epidemiology. Although tools already exist for drawing these diagrams and solving the systems of differential equations they describe, we have created a new software package called StockFlow which uses ideas from category theory to overcome some limitations of existing software. We illustrate this with code in StockFlow that implements a simplified version of a COVID-19 model used in Canada. This is joint work with Xiaoyan Li, Sophie Libkind, Nathaniel Osgood and Evan Patterson.

• ## Past Seminars

• Algebraic Geometry on 30 November 2022 at 15:00

Speaker: Cancelled (n/a)

Title: n/a

Abstract: n/a

• Postgraduate on 30 November 2022 at 12:00

Speaker: Nuno Arala Santos (University of Warwick)

Title: Counting Rational Points on Cubic Surfaces

Abstract: A fundamental problem in Diophantine geometry is to understand the asymptotic behaviour of the number of solutions to a Diophantine equation when we impose a boundedness condition on the variables. We will explain some progress in this problem for equations defining cubic surfaces in 3-dimensional space, following Roger Heath-Brown.

• Algebraic Topology on 29 November 2022 at 16:00

Speaker: Florian Naef (Trinity College Dublin)

Title: Relative intersection product, Whitehead-torsion and string topology

Abstract: Given a closed oriented manifold one can define an intersection product on the homology. This can be extended to local coefficient, and further made relative to the diagonal. I will explain how such a relative self-intersection product is not homotopy invariant (in contrast to the ordinary intersection product) and how this is picked up by string topology. Eventually, we will identify the error term with the trace of Whitehead torsion. More precisely, we will extract an invariant from a Poincare embedding of the diagonal (in the sense of J. Klein) that is the trace of (a version of) Reidemeister torsion. This is based on joint work with P. Safronov.

• Algebra on 28 November 2022 at 17:00

Speaker: Rachel Pengelly (University of Birmingham)

Title: TBC

Abstract: TBC

• Number Theory on 28 November 2022 at 15:00

Title: The arithmetic of the adjoint of a weight 1 modular form

Abstract: A conjecture of Darmon, Lauder and Rotger expresses p-adic iterated integrals attached to a pair of weight 1 modular forms (f,g) in terms of p-adic logarithms of certain units attached to f and g. This talk reports a work in progress in which we explain, in the case where f=g, how to interpret this conjecture as a variant of the Gross-Stark conjecture for the adjoint of f. This requires studying the specializations of the congruence module attached to a Hida deformation of f.

• Colloquium on 25 November 2022 at 16:00

Speaker: Cancelled (-)

Title: -

Abstract: -

• Applied Mathematics on 25 November 2022 at 12:00

Speaker: Eric Neiva (Collége de France & CNRS)

Title: Unfitted finite element methods: decoupling the mesh from the geometry

Abstract: The finite element method (FEM) approximates a PDE from a variational formulation of the problem. Its standard formulation requires a mesh fitting to the boundary of the geometry of interest. Yet, for many problems of practical interest, the geometry is so intricate that mesh generation requires frequent and time-consuming manual intervention. Boundary-fitted meshing can be avoided with unfitted or immersed FEMs. The main idea is to embed the geometry in a simple mesh (e.g., a Cartesian grid) and define the discretisation in the cells intersecting the geometry. In this talk, we will describe a novel unfitted FEM that circumvents the classical issue of immersed FEM: ill-conditioning due small cell-to-geometry intersections We will discuss its application to early embryo development in animals.

• Junior Analysis and Probability Seminar on 24 November 2022 at 13:00

Speaker: Giacomo del Nin (University of Warwick)

Title: Isoperimetric shapes in Penrose tilings

Abstract: TBA

• Probability Seminar on 23 November 2022 at 16:00

Speaker: Pierre-Francois Rodriguez (Imperial College London)

Title: Scaling in low-dimensional long-range percolation models

Abstract: The talk will present recent progress towards understanding the critical behavior of dimensional percolation models exhibiting long-range correlations. The results rigorously exhibit the scaling behavior of various observables of interest and are consistent with scaling theory below the upper-critical dimension (expectedly equal to 6).

• Algebraic Geometry on 23 November 2022 at 15:00

Speaker: Jonathan Lai (Imperial)

Title: A Reconciliation of Mutations and Potentials

Abstract: Given a lattice polygon, one can consider the spanning fan to obtain a toric variety. A combinatorial mutation is an operation that takes one polygon to another, which induces a degeneration of one toric variety to the other. One can then attempt to study all toric degenerations of a fixed Fano variety through the study of polygons and their mutations. In another world, a set of algebraic tori can be glued together by birational maps, also called mutations, to form a cluster variety.

In this talk, I will explain a justification coming from mirror symmetry on why these two operations deserve to share the same name (in dimension 2). Given an orbifold del Pezzo surface X, there is a natural cluster variety Y that knows about the polytopes and mutations associated to X. Namely, there is a combinatorial object associated to Y called a scattering diagram, which is a collection of walls inside a vector space. The chambers, which correspond to tori in Y, are precisely the polygons coming from toric degenerations of X. This is based off ongoing joint work with Tim Magee and Ben Wormleighton.

• Postgraduate on 23 November 2022 at 12:00

Speaker: Alexandros Groutides (University of Warwick)

Title: Galois representations attached to elliptic curves and the Open Image Theorem

Abstract: A Galois representation is a homomorphism $\rho:Gal(\bar{K}/K)\longrightarrow Aut(V)$ where $V$ is a finite dimensional vector space or a free module of finite rank. These objects are of great importance in number theory due to their connections with elliptic curves, modular forms and $L$-functions. We will introduce the mod-$\ell$, $\ell$-adic and adelic Galois representations attached to a non-CM elliptic curve and discuss the structure of their image. The $\ell$-adic open image does not a priori imply the adelic open image but as we will see, it all boils down to the surjectivity of the more innocent sounding mod-$\ell$ representation.

• Algebraic Topology on 22 November 2022 at 16:00

Speaker: Irakli Patchkoria (University of Aberdeen)

Title: Morava K-theory of infinite groups and Euler characteristic

Abstract: Given an infinite discrete group G with a finite model for the classifying space for proper actions, one can define the Euler characteristic of G and the orbifold Euler characteristic of G. In this talk we will discuss higher chromatic analogues of these invariants in the sense of stable homotopy theory. We will study the Morava K-theory of G and associated Euler characteristic, and give a character formula for the Lubin-Tate theory of G. This will generalise the results of Hopkins-Kuhn-Ravenel from finite to infinite groups and the K-theoretic results of Adem, Lück and Oliver from chromatic level one to higher chromatic levels. Along the way we will give explicit computations for amalgamated products of finite groups, right angled Coxeter groups and certain special linear groups. This is all joint with Wolfgang Lück and Stefan Schwede.

• Ergodic Theory and Dynamical Systems on 22 November 2022 at 15:30

Speaker: Tim Austin (UCLA)

Title: Positive sofic entropy without relatively Bernoulli factors.

Abstract: The classical Kolmogorov–Sinai entropy is an invariant of probability-preserving transformations. Much of the resulting theory was successfully extended to actions of discrete amenable groups by Ornstein, Weiss and others.
Lewis Bowen’s more recent notion of sofic entropy extends the Kolmogorov–-Sinai definition to actions of sofic groups, a much larger class introduced by Gromov. A range of natural questions concern how entropy and its consequences differ between the sofic setting the amenable one.
After reviewing a special case of sofic entropy for certain free-product groups, this talk will present a new example of an action of such a group. The example has positive sofic entropy, but has no splitting as a direct product involving a Bernoulli factor. This contrasts with the world of amenable group actions, where many such splittings are guaranteed by the weak Pinsker theorem. The new example is an algebraic action, and its analysis depends on (slight modifications of) results
from the theory of random regular low-density parity-check codes.
This material is part of an ongoing joint project with Lewis Bowen, Brandon Seward and Christopher Shriver.
(This talk will be a continuation of the colloquium from Friday Nov 18th, and will assume some of the notions from that talk.)

• Ergodic Theory and Dynamical Systems on 22 November 2022 at 14:00

Speaker: Timothée Bénard (Cambridge)

Title: The local limit theorem for biased random walks on nilpotent groups

Abstract: We prove the local limit theorem for biased random walks on a simply connected nilpotent Lie group G. The result allows to approximate at scale 1 the n-step distribution of a walk by the time n of a smooth diffusion process for a new group structure on G. We also show this approximation is robust under deviation. The proof uses a Gaussian replacement scheme, combining Fourier analysis and a swapping argument inspired by the work of Diaconis-Hough. As a consequence, we obtain a probabilistic version of Ratner's theorem: Ad-unipotent random walks on finite-volume homogeneous spaces equidistribute toward algebraic measures.

• Partial Differential Equations and their Applications on 22 November 2022 at 12:00

Speaker: Annika Bach (Sapienza Università di Roma)

Title: TBA

Abstract: TBA

• Algebra on 21 November 2022 at 17:00

Speaker: Diego Martin Duro (University of Warwick)

Title: TBC

Abstract: TBC

• Number Theory on 21 November 2022 at 15:00

Speaker: Rachel Greenfeld (IAS)

Title: Aperiodicity of translational tilings

Abstract: Translational tiling is a covering of a space using translated copies of some building blocks, called the "tiles", without any positive measure overlaps. What are the possible ways that a space can be tiled?

A well known conjecture in this area is the periodic tiling conjecture, which asserts that any tile of Euclidean space admits a periodic tiling. In a joint work with Terence Tao, we construct a counterexample to this conjecture. In the talk, I will survey the study of the periodicity of tilings and discuss our recent progress.

• Colloquium on 18 November 2022 at 16:00

Speaker: Tim Austin (UCLA)

Title: Some recent developments around entropy in ergodic theory

Abstract: The entropy rate of a stationary sequence of random symbols was introduced by Shannon in his foundational work on information theory in 1948. In the early 1950s, Kolmogorov and Sinai realized that they could turn this quantity into an isomorphism invariant for measure-preserving transformations on a probability space. Almost immediately, they used it to distinguish many examples called "Bernoulli shifts" up to isomorphism. This resolved a famous open question of the time, and ushered in a new era for ergodic theory.

In the decades since, entropy has become one of the central concerns of ergodic theory, having widespread consequences both for the abstract structure of measure-preserving transformations and for their behaviour in applications. In this talk, I will review some of the highlights of the structural story, and then discuss Bowen's more recent notion of sofic entropy'. This generalizes Kolmogorov--Sinai entropy to measure-preserving actions of many large' non-amenable groups including free groups. I will end with a recent result illustrating how the theory of sofic entropy has some striking differences from its older counterpart.

This talk will be aimed at a general mathematical audience. Most of it should be accessible given a basic knowledge of measure theory, probability, and a little abstract algebra.

• Combinatorics on 18 November 2022 at 14:00

Speaker: Shoham Letzter (UCL)

Title: Separating paths systems of almost linear size

Abstract: A separating path system for a graph G is a collection P of paths in G such that for every two edges e and f, there is a path in P that contains e but not f. We show that every n-vertex graph has a separating path system of size O(n log* n). This improves upon the previous best upper bound of O(n log n), and makes progress towards a conjecture of Falgas-Ravry–Kittipassorn–Korándi–Letzter–Narayanan and Balogh–Csaba–Martin–Pluhár, according to which an O(n) bound should hold.

• Applied Mathematics on 18 November 2022 at 12:00

Speaker: Anna Song (Imperial)

Title: Describing tubular shapes and branching membranes with geometry and topology

Abstract: Tubular and membranous shapes are important mathematical structures that arise in many biomedical applications. Morphology is linked to function: their branching patterns, shaped by interactions and remodelled by diseases, inform us on a biological system.

I will present the “curvatubes” model, which unifies a wide continuum of porous shapes within a common geometric framework (https://doi.org/10.1007/s10851-021-01049-9). It generalizes the Helfrich model for biomembranes by considering shapes as optimizers of a curvature functional in which the principal curvatures may play asymmetric roles. The geometric problem is approximated by a novel phase-field formulation that satisfies a Gamma-limsup property, and is readily implementable as a GPU algorithm. The framework is very flexible and shape textures can be aligned, spatialized, or constrained on a domain.

In the remaining time, I will introduce some topological approaches to analyze such structures using persistent homology, and how they may empirically quantify the "texture of shapes". These are tested on proprietary images of bone marrow vasculature remodelled in acute myeloid leukaemia.

Overall, these compact descriptions offer a unified view to branching tubules and membranes, and will potentially lead to applications in bioengineering, imaging, or materials science.

• Geometry and Topology on 17 November 2022 at 14:00

Speaker: Bradley Zykoski (Univeristy of Michigan)

Title: A polytopal decomposition of strata of translation surfaces

Abstract: A closed surface can be endowed with a certain locally Euclidean metric structure called a translation surface. Moduli spaces that parametrize such structures are called strata. There is a GL(2,R)-action on strata, and orbit closures of this action are rare gems, the classification of which has been given a huge boost in the past decade by landmark results such as the "Magic Wand" theorem of Eskin-Mirzakhani-Mohammadi and the Cylinder Deformation theorem of Wright. Investigation of the topology of strata is still in its nascency, although recent work of Calderon-Salter and Costantini-Möller-Zachhuber indicate that this field is rapidly blossoming. In this talk, I will discuss a way of decomposing strata into finitely many higher-dimensional polytopes. I will discuss how I have used this decomposition to study the topology of strata, and my ongoing work using this decomposition to study the orbit closures of the GL(2,R)-action.

• Junior Analysis and Probability Seminar on 17 November 2022 at 13:00

Speaker: Jakub Takác (University of Warwick)

Title: Norms in finite dimensions and rectifiability in metric spaces

Abstract: TBA

• Probability Seminar on 16 November 2022 at 16:00

Speaker: Sam Olesker-Taylor (University of Warwick)

Title: Random Walks on Random Cayley GraphsTBA

Join conversation
teams.microsoft.com

We investigate mixing properties of RWs on random Cayley graphs of a finite group G with
k≫ 1 independent, uniformly random generators, with 1 ≪ log k ≪ log |G|.

Aldous and Diaconis (1985) conjectured that the RW on this random graph exhibits cutoff for any group G whenever k ≫ log |G| and further that the cutoff time depends only on k and |G|. It was established for Abelian groups.

We disprove the second part of the conjecture by considering RWs on upper-triangular matrices. We extend this conjecture to 1 ≪ k ≲ log |G|, verifying a version of it for arbitrary Abelian groups under 'almost necessary' conditions on k.

It is all joint work with Jonathan Hermon (now at UBC).

• Algebraic Geometry on 16 November 2022 at 15:00

Speaker: Ruijie Yang (Humboldt-Universität zu Berlin)

Title: Zeroes of one forms and homologically trivial deformations

Abstract: In 1926, Hopf proved the Poincaré-Hopf theorem, which implies that if a compact differential manifold admits a nowhere vanishing vector field, then its topological Euler characteristic is zero. Dually, it is natural to ask the same question for one forms. In 1970, Tischler proved that the existence of a nowhere vanishing real closed one form induces a differentiable fiber bundle structure over the circle. In 2013, Kotschick conjectured that for compact Kähler manifolds, admitting a nowhere vanishing real closed one form is actually equivalent to the existence of a nowhere vanishing holomorphic one form. In this talk, I will show that Kotschick’s conjecture can be deduced from a conjecture of Bobadilla-Kollár on homologically trivial deformation. Therefore, Kotschick’s conjecture is true if the first Betti number of X is at least 2dim(X)-2 and the Albanese variety of X is simple. This is joint work with Stefan Schreieder.

• Postgraduate on 16 November 2022 at 12:00

Speaker: Andrew Ronan (University of Warwick)

Title: Exact couples and nilpotent spaces

Abstract: We will introduce spectral sequences via exact couples and outline how to derive the Serre spectral sequence from algebraic topology. Then, we will introduce nilpotent spaces, which are a type of space in many ways dual to a CW complex, before explaining how the Serre spectral sequence can be used to derive some of their properties. For example, the homology groups of a nilpotent space are finitely generated if and only if its homotopy groups are finitely generated.

• Algebraic Topology on 15 November 2022 at 16:00

Speaker: Foling Zou (University of Michigan)

Title: Nonabelian Poincare duality theorem in equivariant factorization homology

Abstract: The factorization homology are invariants of n-dimensional manifolds with some fixed tangential structures that take coefficients in suitable En-algebras. In this talk, I will give a definition for the equivariant factorization homology of a framed manifold for a finite group G by monadic bar construction following Kupers--Miller. Then I will prove the equivariant nonabelian Poincare duality theorem in this case. As an application, in joint work with Asaf Horev and Inbar Klang, we compute the equivariant factorization homology on equivariant spheres for certain Thom spectra. In particular, we recover a computation of the Real topological Hoschild homology by Dotto--Moi--Patchkoria--Reeh.

• Ergodic Theory and Dynamical Systems on 15 November 2022 at 14:00

Speaker: Mar Giralt (Universitat Politecnica de Catalunya)

Title: Chaotic dynamics, exponentially small phenomena and Celestial Mechanics

Abstract: A fundamental problem in dynamical systems is to prove that a given model has chaotic dynamics. One of the methods employed to prove this type of motions is to verify the existence of transversal intersections between the stable and unstable manifolds of certain objects. Then, there exists a theorem (the Smale-Birkhoff homoclinic theorem) which ensures the existence of chaotic motions. In this talk we present a method to analyze the distance and transversality between certain stable and unstable manifolds when a small perturbation is added to an integrable system. In particular, we consider the case where the distance between manifolds is exponentially small. This implies that this difference cannot be detected by expanding the manifolds into a series with respect to the small perturbation parameter. Therefore, classical perturbation theory cannot be applied. Finally, we apply these techniques to a celestial mechanics problem. In particular, we study the Lagrange point L_3 in the restricted planar circular 3-body problem.

• Algebra on 14 November 2022 at 17:00

Speaker: Jay Taylor (University of Manchester)

Title: TBC

Abstract: TBC

• Number Theory on 14 November 2022 at 15:00

Speaker: Rosa Winter (KCL)

Title: Density of rational points on del Pezzo surfaces of degree 1

Abstract: Let X be an algebraic variety over an infinite field k. In arithmetic geometry we are interested in the set X(k) of k-rational points on X. For example, is X(k) empty or not? And if it is not empty, is X(k) dense in X with respect to the Zariski topology?

Del Pezzo surfaces are surfaces classified by their degree d, which is an integer between 1 and 9 (for d >= 3, these are the smooth surfaces of degree d in P^d). For del Pezzo surfaces of degree at least 2 over a field k, we know that the set of k-rational points is Zariski dense provided that the surface has one k-rational point to start with (that lies outside a specific subset of the surface for degree 2). However, for del Pezzo surfaces of degree 1 over a field k, even though we know that they always contain at least one k-rational point, we do not know if the set of k-rational points is Zariski dense in general.

I will talk about density of rational points on del Pezzo surfaces, state what is known so far, and show a result that is joint work with Julie Desjardins, in which we give sufficient and necessary conditions for the set of k-rational points on a specific family of del Pezzo surfaces of degree 1 to be Zariski dense, where k is finitely generated over Q.

• Colloquium on 11 November 2022 at 16:00

Speaker: Yan Fyodorov (King's College London)

Title: "Escaping the crowds": extreme values and outliers in rank-1 non-normal deformations of GUE/CUE

Abstract: Rank-1 non-normal deformations of GUE/CUE provide the simplest model for describing resonances in a quantum chaotic system decaying via a single open channel. In the case of GUE we provide a detailed description of an abrupt restructuring of the resonance density in the complex plane as the function of channel coupling, identify the critical scaling of typical extreme values, and finally describe how an atypically broad resonance (an outlier) emerges from the crowd. In the case of CUE we are further able to study the Extreme Value Statistics of the ''widest resonances'' and find that in the critical regime it is described by a distribution nontrivially interpolating between Gumbel and Frechet. The presentation will be based on the joint works with Boris Khoruzhenko and Mihail Poplavskyi.

• Combinatorics on 11 November 2022 at 14:00

Speaker: Adva Mond (University of Cambridge)

Title: Minimum degree edge-disjoint Hamilton cycles in random directed graphs

Abstract: At most how many edge-disjoint Hamilton cycles does a given directed graph contain? A trivial upper bound is the minimum between the minimum out- and in-degrees. We show that a typical random directed graph D(n,p) contains precisely this many edge-disjoint Hamilton cycles, given that p >= (log^3 n)/n, which is optimal up to a factor of log^

• Applied Mathematics on 11 November 2022 at 12:00

Speaker: Fabian Spill - POSTPONED TO TERM 2 (Birmingham)

Title: Mechanics, Geometry and Topology of Health and Disease

Abstract: TBAExperimental biologists traditionally study biological functions as well as diseases mostly through their abnormal molecular or cellular features. For example, they investigate genetic abnormalities in cancer, hormonal imbalances in diabetes, or an aberrant immune system in vascular diseases. However, many diseases also have a mechanical component which is critical to their deadliness. Notably, cancer kills mostly through metastasis, where the cancer cells acquire the capability to change their physical attachments and migrate. Such mechanical alterations also change geometrical features, such as the cell shape, or topological features, such as the organisation of vascular networks and cellular neighbourhoods within a tissue.

While some of these mechanical, geometrical or topological features in biology are long known, the traditional perspective is to consider them as emergent from molecular features. However, mechanical, geometrical and topological features can also affect the molecular state of a cell. Therefore, the most complete view of many biological systems is to consider them as a complex mechano-chemical systems. Diseases such as cancer are then interpreted as perturbations to this system that cannot be solely explained by considering one feature in isolation (such as a single mutation that ‘causes’ cancer).

I will discuss several examples of systems where this mechanical/geometrical/topological coupling to molecular features plays a crucial role: cells that change their shape, blood vessel cells that open gaps to let cancer cells pass during metastasis, and mitochondria that change their organisation in diabetes.

• Mathematics Teaching and Learning on 10 November 2022 at 16:00

Speaker: Edmund Robertson (St. Andrews)

Title: MacTutor – a collection of great mathematicians?

Abstract: In my talk I will look at questions such as: Is MacTutor a collection of great mathematicians? What is a “great mathematician?” How did I choose whom to write about?

• Junior Analysis and Probability Seminar on 10 November 2022 at 13:00

Speaker: Simon Gabriel (University of Warwick)

Title: On the Poisson–Dirichlet diffusion and Trotter–Kurtz approximations

Abstract: TBA

• Probability Seminar on 09 November 2022 at 16:00

Speaker: Kevin Yang (UC Berkeley)

Title: Time-dependent KPZ equation from non-equilibrium Ginzburg-Landau SDEs

Abstract: This talk has two goals. The first is the derivation of a time-dependent KPZ equation (TDKPZ) from a time-inhomogeneous Ginzburg-Landau model. To our knowledge, said TDKPZ has not yet been derived from microscopic considerations. It has a nonlinear twist that is not seen in the usual KPZ equation, making it a more interesting SPDE.

The second goal is the universality of the method (for deriving TDKPZ), which should work beyond Ginzburg-Landau. In particular, we answer a question of deriving (TD)KPZ from asymmetric particle systems under natural fluctuation-scale versions of the assumptions in Yau’s relative entropy method and a log-Sobolev inequality. This gives some progress on open questions posed at a workshop on KPZ at the American Institute of Math. Time permitting, future directions (of both pure and applied mathematical flavors) will be discussed.

• Algebraic Geometry on 09 November 2022 at 15:00

Speaker: Beihui Yuan (Swansea)

Title: 16 Betti diagrams of Gorenstein Calabi-Yau varieties and a Betti stratification of Quaternary Quartic Forms

Abstract: Motivated by the question of finding all possible projectively normal Calabi-Yau 3-folds in 7-dimensional projective spaces, we proved that there are 16 possible Betti diagrams for arithmetically Gorenstein ideals with regularity 4 and codimension 4. Among them, 8 Betti diagrams have been identified with those of Calabi-Yau 3-folds appeared in a list of 11 families founded by Coughlan-Golebiowski-Kapustka-Kapustka. Another 8 cannot be Betti diagrams of any smooth irreducible nondegenerate 3-fold. Based on the apolarity correspondence between Gorenstein ideals and homogeneous polynomials, and on our results on 16 Betti diagrams, we describe a stratification of the space of quartic forms in four variables.

This talk is based on the paper “Calabi-Yau threefolds in P^n and Gorenstein rings” by Hal Schenck, Mike Stillman and Beihui Yuan, and the preprint “Quaternary quartic forms and Gorenstein rings” by Michal and Grzegorz Kapustka, Kristian Ranestad, Hal Schenck, Mike Stillman and Beihui Yuan.

• Algebraic Topology on 08 November 2022 at 16:00

Speaker: Thibault Décoppet (Oxford)

Title: Fusion 2-Categories associated to 2-groups

Abstract: Motivated by the cobordism hypothesis, which provides a correspondence between fully dualizable objects and fully extended framed TQFTs, it is natural to seek out interesting examples of fully dualizable objects. In dimension four, the fusion 2-categories associated to 2-groups are examples of fully dualizable objects. In my talk, I will begin by reviewing the 2-categorical notion of Cauchy completion, and recall the definition of a fusion 2-category in detail. Then, I will explain how one can construct a fusion 2-category of 2-vector spaces graded by 2-group, and how this construction can be twisted using a 4-cocycle. Finally, it is important to understand when two such fusion 2-categories yield equivalent TQFTs. The answer is provided by the notion of Morita equivalence between fusion 2-categories, which will be illustrated using some examples.

• Number Theory on 08 November 2022 at 15:00

Speaker: George Boxer (Imperial)

Title: Higher Hida theory for Siegel modular varieties

Abstract: The goal of higher Hida theory is to study the ordinary part of coherent cohomology of Shimura varieties integrally.  We introduce a higher coherent cohomological analog of Hida's space of ordinary p-adic modular forms, which is defined as the ordinary part of the coherent cohomology with "partial compact support" of the ordinary Igusa variety. Then we give an analog of Hida's classicality theorem in this setting.  This is joint work with Vincent Pilloni.

• Ergodic Theory and Dynamical Systems on 08 November 2022 at 14:00

Speaker: Donald Robertson (University of Manchester)

Title: Dynamical Cubes and Ergodic Theory

Abstract: n recent works Kra, Moreira, Richter and I showed that positive density sets always contain sums of any finite number of infinite sets, and a shift of the self-sum of an infinite set. The main step in our approach was to prove the existence of certain dynamical configurations described via limit points of orbits. In this talk I will describe what these configurations are, and explain how ergodic theory can be used to deduce their existence.

• Partial Differential Equations and their Applications on 08 November 2022 at 12:00

Speaker: Angeliki Menegaki (IHES)

Title: TBA

Abstract: TBA

• Algebra on 07 November 2022 at 17:00

Speaker: Michael Bate (University of York)

Title: TBC

Abstract: TBC

• Colloquium on 04 November 2022 at 16:00

Speaker: Giovanni Alberti (Pisa)

Title: Small sets in Geometric Measure Theory and Analysis

Abstract: Many relevant problems in Geometric Measure Theory can be ultimately understood in terms of the structure of certain classes of sets, which can be loosely described as "small" (in some sense or another). In this talk I will review a few of these problems and related results, and highlight the connections to other areas of Analysis.

• Combinatorics on 04 November 2022 at 14:00

Speaker: Alp Müyesser (UCL)

Title: Hypergraphs defined by groups

Abstract: This talk will be about a genre of problems where one looks for spanning structures in hypergraphs where vertices represent group elements, and edges represent solutions to systems of equations. Problems expressible using this framework include the Hall-Paige conjecture, the n-queens problem, the harmonious labelling conjecture, Snevily's subsquare conjecture, and many others. We will discuss an absorption-based attack on problems of this type which has resolved many longstanding conjectures in the area.

Joint work with Alexey Pokrovskiy.

• Applied Mathematics on 04 November 2022 at 12:30

Speaker: Emma Davis (Warwick)

Title: Using compartmental ODE models to forecast the elimination of macro-parasitic diseases

Abstract: Standard compartmental models of infectious disease transmission work by categorising a population by stage of infection and then building a system of differential equations that govern the density or number of individuals in each category, e.g. the SIR model has compartments for susceptible (S), infectious (I) and recovered/removed (R) individuals. This makes sense when we are interested in the number of individuals infected over an epidemic where infection is a binary state, as measured by prevalence and incidence, but is less useful for macro-parasitic diseases, where infection is instead classified by the number of macro-parasites inhabiting any given individual. Models for macro-parasitic diseases therefore more commonly consider the number of parasites per individual (their parasite “burden”) or, on a population scale, the mean parasite burden. Common biological features of macro-parasitic diseases, such as sexual reproduction of the parasites or indirect transmission routes, and aggregation between individuals, can result in interesting dynamics at low prevalence, which I will discuss using the example of the macro-parasitic disease lymphatic filariasis.

• Applied Mathematics on 04 November 2022 at 12:00

Speaker: Christian Vaquero-Stainer (Warwick)

Title: The sedimentation dynamics of thin, rigid disks

Abstract: Sedimentation problems arise in a wide range of natural and industrial processes and exhibit a rich array of phenomenology. A particular motivation for this study is the size segregation of graphene flakes, for which a dominant method is centrifugation in a viscous fluid (Khan 2012). We present a numerical investigation of the sedimentation dynamics of thin, deformed circular disks sedimenting freely under gravity in an otherwise quiescent, Stokesian fluid. In the first part of this study, we address singularities which arise in the fluid pressure and velocity gradient at the edge of the disk, by developing an augmented finite element method to capture the singularities with analytic functions. In the second part of the study, we deploy this method in a fluid-structure interaction framework to examine the behaviour of two distinct classes of disk shape, namely cylindrically- and conically-deformed disks with one and two planes of symmetry, respectively. We explore the geometry-driven dynamics and the bifurcation structure which arises for the conically-deformed disk as the level of asymmetry is varied.

• Geometry and Topology on 03 November 2022 at 14:00

Speaker: Becca Winarski (College of the Holy Cross)

Title: Polynomials, branched covers, and trees

Abstract: Thurston proved that a post-critically finite branched cover of the plane is either equivalent to a polynomial (that is: conjugate via a mapping class) or it has a topological obstruction. We use topological techniques – adapting tools used to study mapping class groups – to produce an algorithm that determines when a branched cover is equivalent to a polynomial, and if it is, determines which polynomial a topological branched cover is equivalent to. This is joint work with Jim Belk, Justin Lanier, and Dan Margalit.

• Probability Seminar on 02 November 2022 at 16:00

Speaker: Alessandra Cipriani (University College London)

Title: Properties of the gradient squared of the Gaussian free field

Abstract: TBA

• Algebraic Geometry on 02 November 2022 at 15:00

Speaker: Michel van Garrel (Birmingham)

Title: Log Mirror Symmetry

Abstract: Start with a smooth Fano variety X and a smooth anticanonical divisor D. Consider the problem of counting maps from the projective line to X that meet D along a curve in only one point. While this problem is intractable directly, in this joint work with Helge Ruddat and Bernd Siebert, we use toric dualities to translate the problem into a dual problem in a dual geometry. There the problem turns into a problem of computing period integrals, which we can readily solved via the techniques of Picard-Fuchs equations.

In my talk, I will limit to the case of X the projective plane and D a smooth conic. The before-mentioned toric dualities are the constructions of the Gross-Siebert programme. I hope to convey the observation that the dualities are natural and that the translation from counting problem to period integral is as well.

• Ergodic Theory and Dynamical Systems on 01 November 2022 at 14:00

Speaker: Carlos Matheus (Ecole Polytechnique)

Title: Elliptic dynamics on certain SU(2) and SU(3) character varieties

Abstract: In this talk, we discuss the action of a hyperbolic element of SL(2,Z) on the SU(2) and SU(3) character varieties of once-punctured torii. This is based on a joint work with G. Forni, W. Goldman and S. Lawton.

• Algebra on 31 October 2022 at 17:00

Speaker: Sean Eberhard (University of Cambridge)

Title: The Boston--Shalev conjecture for conjugacy classes

Abstract: The Boston--Shalev conjecture (proved by Fulman and Guralnick in 2015) asserts that in any nonabelian simple group G in any nontrivial permutation action the proportion of derangements is at least some absolute constant c > 0. Since the set of derangements is closed under conjugacy it is also natural to ask about the proportion of *conjugacy classes* containing derangements. It is easy to see that this version of the question has a negative answer for alternating groups, but Guralnick and Zalesski asked whether it holds for groups of Lie type. I will outline a proof. We can also (1) extend to the case of almost simple groups, which is not true for the original conjecture, and (2) deduce the original conjecture, which amounts to a simplification of the Fulman--Guralnick proof. The key turns out to be a kind of analytic number theory for palindromic polynomials. This is ongoing work with Daniele Garzoni.

• Number Theory on 31 October 2022 at 15:00

Speaker: Yoav Gath (Cambridge)

Title: Lattice point statistics for Cygan–Koranyi balls

Abstract: Euclidean lattice point counting problems, the classical example of which is the Gauss circle problem, are an important topic in classical analysis and have been the driving force behind much of the developments in the area of analytic number theory in the 20th century. In this talk, I will introduce the lattice point counting problem for (2q+1)-dimensional Cygan–Koranyi balls, namely, the problem of establishing error estimates for the number of integer lattice points lying inside Heisenberg dilates of the unit ball with respect to the Cygan–Koranyi norm. I will explain how this problem arises naturally in the context of the Heisenberg groups, and how it relates to the Euclidean case (and in particular to the Gauss circle problem). I will survey some of the major results obtained to date for this lattice point counting problem, and in particular, results related to the fluctuating nature of the error term.

• Colloquium on 28 October 2022 at 16:00

Speaker: Tom Gur (Warwick Computer Science)

Title: Quantum algorithms and additive combinatorics

Abstract: I will discuss a new connection between quantum computing and additive combinatorics, which allows for boosting the power of quantum algorithms. Namely, I will show a framework that uses generalisations of Bogolyubov’s lemma and Sander’s quasi-polynomial Bogolyubov-Ruzsa lemma to transform quantum algorithms that are only correct on a small number of inputs into quantum algorithms that are correct on all inputs.

• Applied Mathematics on 28 October 2022 at 12:00

Speaker: Maciej Buze (Birmingham)

Title: Mathematical analysis of atomistic fracture and related phenomena in crystalline materials

Abstract: The modelling of atomistic fracture and related phenomena in crystalline materials poses a string of mathematically non-trivial and exciting challenges, both on the theoretical and practical level. At the heart of the problem lies a discrete domain of atoms (a lattice), which exhibits spatial inhomogeneity induced by the crack surface, particularly pronounced in the vicinity of the crack tip. Atoms interact in a highly nonlinear way, resulting in a severely non-convex energy landscape facilitating non-trivial behaviour of atoms such as (i) crack propagation; (ii) near-crack tip plasticity - emission and movement of defects known as dislocations in the vicinity of the crack tip; (iii) surface effects - atoms at the crack surface relaxing or possibly attaining an altogether different crystalline structure. On the practical side, the richness of possible phenomena renders the task of setting up numerical simulations particularly tricky - numerical artefacts, e.g. induced by prescribing a particular boundary condition, can lead to inconsistent results. In this talk I will aim to summarise on-going efforts aimed at putting the atomistic modelling of fracture on a rigorous mathematical footing. I will introduce a framework giving rise to well-defined models for which regularity and stability of solutions can be discussed (topic of my PhD thesis at Warwick). I will then show how the theory can be used to set up practical simulations, such as Mode I fracture of silicon on the (111) cleavage plane using state-of-the-art interatomic potentials. I will also outline how this framework can be used to rigorously derive upscaled models of near-crack-tip plasticity. Finally, I will also talk about challenges in addressing the surface effects.

• Geometry and Topology on 27 October 2022 at 14:00

Speaker: Daniel Berlyne (University of Bristol)

Title: Braid groups of graphs

Abstract: The braid group of a space X is the fundamental group of its configuration space, which tracks the motion of some number of particles as they travel through X. When X is a graph, the configuration space turns out to be a special cube complex, in the sense of Haglund and Wise. I show how these cube complexes are constructed and use graph of groups decompositions to provide methods for computing braid groups of various graphs, as well as criteria for a graph braid group to split as a free product. This has various applications, such as characterising various forms of hyperbolicity in graph braid groups and determining when a graph braid group is isomorphic to a right-angled Artin group.

• Algebraic Geometry on 26 October 2022 at 15:00

Speaker: Arman Sarikyan (Edinburgh)

Title: On the Rationality of Fano-Enriques Threefolds

Abstract: A three-dimensional non-Gorenstein Fano variety with at most canonical singularities is called a Fano-Enriques threefold if it contains an ample Cartier divisor that is an Enriques surface with at most canonical singularities. There is no complete classification of Fano-Enriques threefolds yet. However, L. Bayle has classified Fano-Enriques threefolds with terminal cyclic quotient singularities in terms of their canonical coverings, which are smooth Fano threefolds in this case. The rationality of Fano-Enriques threefolds is an open classical problem that goes back to the works of G. Fano and F. Enriques. In this talk we will discuss the rationality of Fano-Enriques threefolds with terminal cyclic quotient singularities.

• Postgraduate on 26 October 2022 at 12:00

Speaker: Robin Visser (University of Warwick)

Title: Hilbert's Tenth Problem

Abstract: Can you find four distinct positive integers $w,x,y,z$ such that $w^3+x^3=y^3+z^3$ ?

If that's too easy, try finding a non-trivial integer solution to $x^4+y^4+z^4=w^4$.

And good luck finding any integral solution to $x^3+y^3+z^3=114$.

This all begs the question of whether we can construct a general algorithm to determine whether any given Diophantine equation has integer solutions. David Hilbert posed this exact question at the second ICM in 1900, where a negative answer was finally proven 70 years later by Yuri Matiyasevich building on work by Martin Davis, Hilary Putnam and Julia Robinson. In this talk, we'll explore the mathematical ideas behind Hilbert's tenth problem as well as go over many surprising applications, extensions to other number fields, and how this relates to several other famous open problems!

• Algebraic Topology on 25 October 2022 at 16:00

Speaker: Severin Bunk (Oxford)

Title: Functorial field theories from differential cocycles

Abstract: In this talk I will demonstrate how differential cocycles give rise to (bordism-type) functorial field theories (FFTs). I will discuss some background on smooth FFTs, differential cohomology and higher gerbes with connection as a geometric model for differential cocycles before explaining the general principle for how to obtain smooth FFTs from higher gerbes. In the second part, I will focus on the two-dimensional case. Here I will present a concrete, geometric construction of two-dimensional smooth FFTs on background manifolds, starting from gerbes with connection. This is related to WZW theories. If time permits, I will comment on an extension of this construction which produces open-closed field theories.

• Ergodic Theory and Dynamical Systems on 25 October 2022 at 14:00

Speaker: Maryam Hosseini (Open University)

Title: About Minimal Dynamics on the Cantor Set

Abstract: Dimension group is an operator algebraic object related to minimal dynamical systems on the Cantor set. In this talk after a quick review of some definitions of dimension group, the {\it topological and algebraic rank} of Cantor minimal systems are considered and we will see how the rank of a Cantor system is dominated by the rank of its extensions.

• Partial Differential Equations and their Applications on 25 October 2022 at 12:00

Speaker: Alexandra Tzella (University of Birmingham)

Title: TBA

Abstract: TBA

• Algebra on 24 October 2022 at 17:00

Speaker: Alexandre Zalesski (UEA)

Title: Some problems on representations of simple algebraic groups

Abstract: Some open questions on the weight structure of tensor-decomposable representations of simple algebraic groups will be discussed.

• Number Theory on 24 October 2022 at 15:00

Speaker: Aleksander Horawa (Oxford)

Title: Motivic action on coherent cohomology of Hilbert modular varieties

Abstract: A surprising property of the cohomology of locally symmetric spaces is that Hecke operators can act on multiple cohomological degrees with the same eigenvalues. We will discuss this phenomenon for the coherent cohomology of line bundles on modular curves and, more generally, Hilbert modular varieties. We propose an arithmetic explanation: a hidden degree-shifting action of a certain motivic cohomology group (the Stark unit group). This extends the conjectures of Venkatesh, Prasanna, and Harris to Hilbert modular varieties.

• Colloquium on 21 October 2022 at 16:00

Speaker: Pierre Raphael (Cambridge)

Title: Singularity formation for super critical waves

Abstract: Give a wave packet an initial energy and let it propagate in the whole space, then in the linear regime, the wave packet will scatter. But in non linear regimes, part of the energy may concentrate to form coherent non linear structures which propagate without deformation (solitons). And in more extreme cases singularities may form. Whether or not singular structures arise is a delicate problem which has attracted a considerable amount of works in both mathematics and physics, in particular in the super critical regime which is the heart of the 6Th Clay problem on singularity formation for three dimensional viscous incompressible fluids. For another classical model like the defocusing Non Linear Schrodinger equation (NLS), Bourgain ruled out in a breakthrough work (1994) the existence of singularities in the critical case, and conjectured that this should extend to the super critical one. I will explain how the recent series of joint works with Merle (IHES), Rodnianski (Princeton) and Szeftel (Paris 6) shades a new light on super critical singularities: in fact there exist super critical singularities for (NLS), and the new underlying mechanism is directly connected to the first description of singularities for three dimensional viscous compressible fluids.

• Combinatorics on 21 October 2022 at 14:00

Speaker: Daniel Iľkovič (Masaryk University)

Title: Quasirandom tournaments

Abstract: A directed graph H is quasirandom-forcing for tournaments if the limit (homomorphic) density of H in a sequence of tournaments is 2^−|E(H)| if and only if the sequence is quasirandom. The cyclic orientation of a cycle of length k is quasirandom-forcing if and only if k = 2 mod 4. We study a generalization of this result: what orientations of a cycle of length k are quasirandom-forcing? We show that no orientation of an odd cycle is quasirandom-forcing and classify which orientations of even cycles of length up to 10 are quasirandom-forcing. This is joint work with Andrzej Grzesik, Bartosz Kielak and Dan Kráľ

• Applied Mathematics on 21 October 2022 at 12:00

Speaker: Francis Aznaran (Oxford)

Title: Finite element methods for the Stokes–Onsager–Stefan–Maxwell equations of multicomponent flow

Abstract: The Onsager framework for linear irreversible thermodynamics provides a thermodynamically consistent model of mass transport in a phase consisting of multiple species, via the Stefan–Maxwell equations, but a complete description of the overall transport problem necessitates also solving the momentum equations for the flow velocity of the medium. We derive a novel nonlinear variational formulation of this coupling, called the (Navier–)Stokes–Onsager–Stefan–Maxwell system, which governs molecular diffusion and convection within a non-ideal, single-phase fluid composed of multiple species, in the regime of low Reynolds number in the steady state. We propose an appropriate Picard linearisation posed in a novel Sobolev space relating to the diffusional driving forces, and prove convergence of a structure-preserving finite element discretisation. This represents some of the first rigorous numerics for the coupling of multicomponent molecular diffusion with compressible convective flow. The broad applicability of our theory is illustrated with simulations of the centrifugal separation of noble gases and the microfluidic mixing of hydrocarbons. This is joint work with Alexander Van-Brunt.

• Geometry and Topology on 20 October 2022 at 14:00

Title: TBA

Abstract: TBA

• Algebraic Geometry on 19 October 2022 at 15:00

Speaker: Nivedita Viswanathan (Loughborough)

Title: On the Rationality of Fano-Enriques Threefolds

Abstract: There has been a lot of development recently in understanding the existence of Kahler-Einstein metrics on Fano manifolds due to the Yau-Tian-Donaldson conjecture, which gives us a way of looking at this problem in terms of the notion of K-stability. In particular, this problem is solved in totality for smooth del Pezzo surfaces by Tian. For del Pezzo surfaces with quotient singularities, there are partial results. In this talk, we will consider singular del Pezzo surfaces which are quasi-smooth, well-formed hypersurfaces in weighted projective space, and understand what we can say about their K-stability. This is ongoing joint work with In-Kyun Kim and Joonyeong Won.

• Postgraduate on 19 October 2022 at 12:00

Speaker: Ruzhen Yang (University of Warwick)

Title: Beilinson spectral sequence and its reverse problems on PP^2

Abstract: Derived category is widely accepted as the natural environment to study homological algebra. We will study the structure of the bounded derived category of coherent sheaves on projective space via the semi-orthogonal decomposition (based on the Beilinson's theorem) and comparison (by a theorem by A. Bondal). As an example, we will give explicit free resolutions of some sheaves on P2 using the Beilinson spectral sequence. We will also discuss the reverse problem where we give a condition to when the complex given by the spectral sequence is a resolution of the ideal sheaf of three points.

• Algebraic Topology on 18 October 2022 at 16:00

Title: G-typical Witt vectors with coefficients and the norm

Abstract: The norm is an important construction on equivariant spectra, most famously playing a key role in the work of Hill, Hopkins and Ravenel on the Kervaire invariant one problem. Witt vectors are an algebraic construction first used in Galois theory in the 1930s, but later finding applications in stable equivariant homotopy theory. I will describe a new generalisation of Witt vectors that can be used to compute the zeroth equivariant stable homotopy groups of the norm $N_e^G Z$, for $G$ a finite group and $Z$ a connective spectrum.

• Ergodic Theory and Dynamical Systems on 18 October 2022 at 14:00

Speaker: Joe Thomas (Durham University)

Title: Poisson statistics, short geodesics and small eigenvalues on hyperbolic punctured spheres

Abstract: For hyperbolic surfaces, there is a deep connection between the geometry of closed geodesics and their spectral theoretic properties. In this talk, I will discuss recent work with Will Hide (Durham), where we study both sides of this relationship for hyperbolic punctured spheres. In particular, we consider Weil-Petersson random surfaces and demonstrate Poisson statistics for counting functions of closed geodesics with lengths on scales 1/sqrt(number of cusps), in the large cusp regime. Using similar ideas, we show that typical hyperbolic punctured spheres with many cusps have lots of arbitrarily small eigenvalues. Throughout, I will contrast these findings to the setting of closed hyperbolic surfaces in the large genus regime.

• Algebra on 17 October 2022 at 17:00

Speaker: Kamilla Rekvényi (Imperial College London)

Title: The Orbital Diameter of Primitive Permutation Groups

Abstract: Let G be a group acting transitively on a finite set Ω. Then G acts on ΩxΩ component wise. Define the orbitals to be the orbits of G on ΩxΩ. The diagonal orbital is the orbital of the form ∆ = {(α, α)|α ∈ Ω}. The others are called non-diagonal orbitals. Let Γ be a non-diagonal orbital. Define an orbital graph to be the non-directed graph with vertex set Ω and edge set (α,β)∈ Γ with α,β∈ Ω. If the action of G on Ω is primitive, then all non-diagonal orbital graphs are connected. The orbital diameter of a primitive permutation group is the supremum of the diameters of its non-diagonal orbital graphs. There has been a lot of interest in finding bounds on the orbital diameter of primitive permutation groups. In my talk I will outline some important background information and the progress made towards finding explicit bounds on the orbital diameter. In particular, I will discuss some results on the orbital diameter of the groups of simple diagonal type and their connection to the covering number of finite simple groups. I will also discuss some results for affine groups, which provides a nice connection to the representation theory of quasisimple groups.

• Number Theory on 17 October 2022 at 15:00

Title: L-invariants for cohomological representations of PGL(2) over an arbitrary number field

Abstract: In this talk I will construct the automorphic L-invariant attached to a cuspidal representation π of PGL(2) over an arbitrary number field F, and a prime p of F such that the local component πp is the Steinberg representation and π is non-critical at p. I will show that, if F is totally real then the automorphic L-invariant attached to π and p agrees with the derivatives of the Up-eigenvalue of the p-adic family passing through π. From this I will deduce the equality between the automorphic L-invariant and the Fontaine-Mazur L-invariant of the associated Galois representation. This is a joint work with Lennart Gehrmann.

• Colloquium on 14 October 2022 at 16:00

Speaker: David Rand (Warwick)

Title: An appreciation of Christopher Zeeman

Abstract: In writing a biographical memoir of Zeeman for the Royal Society, I appreciated even more what a remarkable character he was, both in terms of his life, his leadership, his mathematics and his breadth of interests. I discovered a number of aspects that I don't think are very well known about his life and his contributions to topology, catastrophe theory and our department. In this colloquium I will try and give an overview of this.

• Combinatorics on 14 October 2022 at 14:00

Speaker: Candy Bowtell (University of Warwick)

Title: The n-queens problem

Abstract: The n-queens problem asks how many ways there are to place n queens on an n x n chessboard so that no two queens can attack one another, and the toroidal n-queens problem asks the same question where the board is considered on the surface of a torus. Let Q(n) denote the number of n-queens configurations on the classical board and T(n) the number of toroidal n-queens configurations. The toroidal problem was first studied in 1918 by Pólya who showed that T(n)>0 if and only if n is not divisible by 2 or 3. Much more recently Luria showed that T(n) is at most ((1+o(1))ne^{-3})^n and conjectured equality when n is not divisible by 2 or 3. We prove this conjecture, prior to which no non-trivial lower bounds were known to hold for all (sufficiently large) n not divisible by 2 or 3. We also show that Q(n) is at least ((1+o(1))ne^{-3})^n for all natural numbers n which was independently proved by Luria and Simkin and, combined with our toroidal result, completely settles a conjecture of Rivin, Vardi and Zimmerman regarding both Q(n) and T(n). In this talk we'll discuss our methods used to prove these results. A crucial element of this is translating the problem to one of counting matchings in a 4-partite 4-uniform hypergraph. Our strategy combines a random greedy algorithm to count almost' configurations with a complex absorbing strategy that uses ideas from the methods of randomised algebraic construction and iterative absorption. This is joint work with Peter Keevash.

• Applied Mathematics on 14 October 2022 at 12:00

Speaker: Philip Herbert (Heriot-Watt)

Title: Shape optimisation with Lipschitz functions

Abstract: In this talk, we discuss a novel method in PDE constrained shape optimisation. We begin by introducing the concept of PDE constrained shape optimisation. While it is known that many shape optimisation problems have a solution, their approximation in a meaningful way is non-trivial. To find a minimiser, it is typical to use first order methods. The novel method we propose is to deform the shape with fields which are a direction of steepest descent in the topology of W^1_\infty. We present an analysis of this in a discrete setting along with the existence of directions of steepest descent. Several numerical experiments will be considered which compare a classical Hilbertian approach to this novel approach.

• Geometry and Topology on 13 October 2022 at 14:00

Speaker: Claudio Llosa Isenrich (KIT)

Title: Finiteness properties, subgroups of hyperbolic groups and complex hyperbolic lattices

Abstract: Hyperbolic groups form an important class of finitely generated groups that has attracted much attention in Geometric Group Theory. We call a group of finiteness type $F_n$ if it has a classifying space with finitely man cells of dimension at most n, generalising finite presentability, which is equivalent to type $F_2$. Hyperbolic groups are of type $F_n$ for all $n$ and it is natural to ask if their subgroups inherit these strong finiteness properties. We use methods from complex geometry to show that every uniform arithmetic lattice with positive first Betti number in $PU(n,1)$ admits a finite index subgroup, which maps onto the integers with kernel of type $F_{n-1}$ and not $F_n$. This answers an old question of Brady and produces many finitely presented non-hyperbolic subgroups of hyperbolic groups. This is joint work with Pierre Py.

• Algebraic Topology on 11 October 2022 at 16:00

Speaker: Sebastian Chenery (University of Southampton)

Title: On Pushout-Pullback Fibrations

Abstract: We will discuss recent work inspired by a paper of Jeffrey and Selick, where they ask whether the pullback bundle over a connected sum can itself be homeomorphic to a connected sum. We provide a framework to tackle this question through classical homotopy theory, before pivoting to rational homotopy theory to give an answer after taking based loop spaces.

• Ergodic Theory and Dynamical Systems on 11 October 2022 at 14:00

Speaker: Sabrina Kombrink (University of Birmingham)

Title: TBA

Abstract: TBA

• Algebra on 10 October 2022 at 17:00

Speaker: Gareth Tracey (University of Warwick)

Title: Primitive amalgams and the Goldschmidt-Sims conjecture

Abstract: The Classification of Finite Simple Groups has led to substantial progress on deriving sharp order bounds in various natural families of finite groups. One of the most well-known instances of this is Sims' conjecture, which states that a point stabiliser in a primitive permutation group has order bounded in terms of its smallest non-trivial orbit length (this was proved by Cameron, Praeger, Saxl and Seitz using the CFSG in 1983). In the meantime, Goldschmidt observed that a generalised version of Sims' conjecture, which we now call the \emph{Goldschmidt--Sims conjecture}, would lead to important applications in graph theory. In this talk, we will describe the conjecture, and discuss some recent progress. Joint work with L. Pyber.

• Number Theory on 10 October 2022 at 15:00

Speaker: Lambert A'Campo (Oxford)

Title: Galois representations and cohomology of congruence subgroups

Abstract: In this talk I will explain what it means to attach Galois representations to the cohomology of arithmetic locally symmetric spaces arising from congruence subgroups. In the case of GL(2) over imaginary CM fields (the method also works for GL(n)) I will explain how to prove, under certain conditions, that the Galois representations constructed by Harris–Lan–Taylor–Thorne and Scholze have good p-adic Hodge theoretic properties.

• DAGGER on 10 October 2022 at 14:00

Speaker: Marco Linton (University of Oxford)

Title: Poison subgroups for hyperbolic groups

Abstract: It is a well-known result that hyperbolic groups cannot contain certain `poison' subgroups. A lot of progress has been made towards understanding when the converse to this statement also holds. This includes several positive results, but also several negative results. In this talk, I will introduce hyperbolic groups, discuss some of these results and present the current state of the art for the class of one-relator groups.

• Colloquium on 07 October 2022 at 16:00

Speaker: Michela Ottobre (Heriot-Watt)

Title: Interacting Particle systems and (Stochastic) Partial Differential equations: modelling, analysis and computation

Abstract: The study of Interacting Particle Systems (IPSs) and related kinetic equations has attracted the interest of the mathematics and physics communities for decades. Such interest is kept alive by the continuous successes of this framework in modelling a vast range of phenomena, in diverse fields such as biology, social sciences, control engineering, economics, game theory, statistical sampling and simulation, neural networks etc. While such a large body of research has undoubtedly produced significant progress over the years, many important questions in this field remain open. We will (partially) survey some of the main research directions in this field and discuss open problems.

• Combinatorics on 07 October 2022 at 14:00

Speaker: Mustazee Rahman (University of Durham)

Title: Suboptimality of local algorithms for optimization on sparse graphs

Abstract: Suppose we want to find the largest independent set or maximal cut in a large yet sparse graph, where the average vertex degree is constant. These are two basic optimization problems relevant to both theory and practice. For typical, or rather random sparse graphs, many algorithms proceed by way of local decision rules. Examples include Glauber dynamics, Belief propagation, etc. I will explain a form of local algorithm that captures many of these. I will then explain how they fail to find optimal independent sets or cuts once the average degree of the graph gets large. Along the way, we will find connections to entropy and spin glasses.

• Geometry and Topology on 06 October 2022 at 14:00

Speaker: Grace Garden (University of Sydney)

Title: Earthquakes on the once-punctured torus

Abstract: We study earthquake deformations on Teichmüller space associated with simple closed curves of the once-punctured torus. We describe two methods to get an explicit form of the earthquake deformation for any simple closed curve. The first method is rooted in hyperbolic geometry, the second representation theory. The two methods align, providing both a geometric and an algebraic interpretation of the earthquake deformations. Pictures are given for earthquakes across multiple coordinate systems for Teichmüller space. Two families of curves are used as examples. Examining the limiting behaviour of each gives insight into earthquakes about measured geodesic laminations, of which simple closed curves are a special case.

• Algebra on 05 October 2022 at 16:17

Speaker: Matija Vidmar (University of Ljubljana)

Title: Noise Boolean algebras: classicality, blackness and spectral independence

Abstract: Informally speaking, a noise Boolean algebra is an aggregate of pieces of information, subject to statistical independence properties relative to an underlying notion of chance. More formally, it is a distributive sublattice of the lattice of all sub-sigma-fields of a given probability space, each element of which admits an independent complement. A noise Boolean algebra is classical (resp. black) when all its random variables are stable (resp. sensitive) under infinitesimal perturbations of its basic ingredients. For instance, the Wiener and Poisson noises are classical, but certain noises of percolation and coalescence are black. We shall see that classicality and blackness are respectively characterized by existence and non-existence of certain so-called spectral independence probabilities that we shall introduce. Associated preprint: https://drive.google.com/file/d/1cLOHpHG_xgqPYmsbVIQmmYx08pqk4H6m/view

• Postgraduate on 05 October 2022 at 12:00

Speaker: Sunny Sood (University of Warwick)

Title: Homological stability for $O_{n,n}$

Abstract: Motivated by Hermitian K-Theory, we study the homological stability of the split orthogonal group $O_{n,n}$. Specifically, let $R$ be a commutative local ring with infinite residue field such that $2 \in R^{*}$. We prove that the natural homomorphism $H_{k}(O_{n,n}(R) ; \mathbb{Z}) \rightarrow H_{k}(O_{n+1,n+1}(R); \mathbb{Z})$ is an isomorphism for $k \leq n-1$ and surjective for $k \leq n$. This will be an excellent opportunity to introduce esoteric concepts such as group homology and hyperhomology spectral sequences at the postgraduate seminar. This is all joint work with my supervisor Dr Marco Schlichting.

• Partial Differential Equations and their Applications on 04 October 2022 at 12:00

Speaker: Tobias Barker (University of Bath)

Title: A quantitative approach to the Navier–Stokes equations

Abstract: ecently, Terence Tao used a new quantitative approach to infer that certain ‘slightly supercritical’ quantities for the Navier–Stokes equations must become unbounded near a potential blow-up time. In this talk I’ll discuss a new strategy for proving quantitative bounds for the Navier–Stokes equations, as well as applications to behaviours of potentially singular solutions. This talk is based upon joint work with Christophe Prange (CNRS, Cergy Paris Université).

• Number Theory on 03 October 2022 at 15:00

Speaker: Matteo Tamiozzo (Warwick)

Title: Perfectoid quaternionic Shimura varieties and the Jacquet–Langlands correspondence

Abstract: The Hodge–Tate period map can be thought of as a p-adic analogue of the Borel embedding. However, unlike its complex counterpart, it is not injective, and the pushforward of the constant sheaf via the Hodge–Tate period map encodes interesting arithmetic information. In the setting of quaternionic Shimura varieties, I will explain the relation between the structure of this complex of sheaves and level raising and the Jacquet–Langlands correspondence. I will then discuss applications to the study of the cohomology of quaternionic Shimura varieties. I will illustrate most of the arguments in the simplest setting of modular and Shimura curves. This is joint work with Ana Caraiani.

• DAGGER on 03 October 2022 at 14:00

Speaker: Aleksi Pyörälä (University of Oulu)

Title: Normal numbers in self-conformal sets

Abstract: During recent years, the prevalence of normal numbers in natural subsets of the reals has been an active research topic in fractal geometry. The general idea is that in the absence of any special arithmetic structure, almost all numbers in a given set should be normal, in every base. In our recent joint work with Balázs Bárány, Antti Käenmäki and Meng Wu we verify this for self-conformal sets on the line. The result is a corollary of a uniform scaling property of self-conformal measures: roughly speaking, a measure is said to be uniformly scaling if the sequence of successive magnifications of the measure equidistributes, at almost every point, for a common distribution supported on the space of measures. Dynamical properties of these distributions often give information on the geometry of the uniformly scaling measure.